On the reaction term in (REF ) we impose that it is locally bounded with respect to \(u\in (0,\infty )\)  and allow for a singularity in zero.
In particular, we ask that \({\Vert f({\,\cdot \,},z)\Vert }_\infty \le Lz^{-m_0}\) , \(L\ge 0\) , for all \(z\in [0,1)\)  for a certain \(m_0>0\)  depending on the structure of the degeneracy of \(\phi \)  in zero.
Motivated by the application to the biofilm growth model and to simplify the analysis, we had first imposed that the reaction term is bounded on the interval \([0,1)\) .
We were able to show interior Hölder continuity under this assumption.
Later, we observed that our proof allowed for even more general reaction terms that might be singular in 0, so we include those in our result.
It differs from standard assumptions in the literature on regularity theory, see [1]}, [2]}, [3]}, [4]}.
We can show that the more general solution concept used in [4]} has a time derivative that can be interpreted as a bounded linear functional and therefore, is included in our solution concept, provided that \(f({\,\cdot \,},u)\in L^2_{\mathrm {loc}}(\Omega \times (0,T))\) .
For details, see Proposition 3.19 in [6]}.
